3-9 Convergence of meridians

Summary

This video delves into the concept of the convergence of meridians in geodesy, explaining the difference between planar, spherical, and ellipsoidal datums. It covers how meridians converge towards the North Pole, affecting direction and distance calculations. The video introduces the concept of spherical triangles and explains how to compute corrections for meridian convergence using spherical trigonometry. The process includes practical steps for calculating azimuths and correcting for the convergence of meridians on both spherical and ellipsoidal surfaces, with an example demonstrating these corrections for a specific ellipsoid and geographic coordinates.

Takeaways

• 😀 The convergence of meridians occurs on a spherical or ellipsoidal datum, where meridians point towards the North and eventually meet at the North Pole.
• 😀 In a planar datum, the coordinate system is defined such that the Y-axis points toward the North, and all directions of North are parallel along any line.
• 😀 Meridians on a spherical or ellipsoidal datum do not remain parallel, but instead converge towards the North Pole, causing a difference in direction to North at different locations.
• 😀 Spherical triangles can be used to model the geometry of the Earth's surface, with points having latitude and longitude, and angles defined relative to the equator.
• 😀 The angle of convergence (gamma) between meridians is important for correcting azimuth calculations on an ellipsoidal surface.
• 😀 The direct and reverse azimuths in planar and spherical datums can be calculated using the angles and distances between points, considering the convergence of meridians.
• 😀 The reverse azimuth differs from the direct azimuth by 180° in planar geometry, but for spherical and ellipsoidal geometry, a convergence angle (gamma) must be considered, leading to a more complex calculation.
• 😀 In spherical geometry, the reverse azimuth can be computed by adding the direct azimuth to pi (or 180°) and correcting for the convergence angle (gamma).
• 😀 The formula for calculating gamma involves spherical trigonometry, specifically using sine laws to relate the angle of convergence to the arc between points.
• 😀 On an ellipsoid, the correction for convergence (gamma) is more complicated due to the ellipsoidal shape, and includes parameters like the radius of the prime vertical and eccentricity.
• 😀 A practical example demonstrates how to compute the correction for convergence on an ellipsoid using the WGS84 datum, resulting in a correction value expressed in degrees, minutes, and seconds.

Q & A

What is meant by the convergence of meridians in surveying?

The convergence of meridians refers to the phenomenon where meridians (lines of longitude) on a spherical or ellipsoidal surface meet at the North and South Poles, unlike parallel lines in a planar coordinate system. This causes the direction of 'North' to vary slightly depending on the location, creating a need for corrections in calculations, such as azimuths.

How does the convergence of meridians affect azimuth calculations?

Azimuths are the angles measured from the North direction to the line connecting two points. In a planar datum, reverse azimuths differ by exactly 180°. However, in spherical and ellipsoidal datums, due to the convergence of meridians, the reverse azimuths deviate slightly. This requires an additional correction angle, denoted as gamma (γ), to accurately compute reverse azimuths.

What is a spherical triangle, and how is it used in surveying?

A spherical triangle is formed on the surface of a sphere, typically by connecting the North Pole and two other points on the sphere’s surface. The sides of the triangle are arcs of great circles, and the angles are measured at the sphere's center. In surveying, these triangles are used to compute distances and angles between points on a spherical or ellipsoidal datum, particularly for determining azimuths.

Why is spherical trigonometry important in calculating azimuths on a spherical or ellipsoidal datum?

Spherical trigonometry is crucial because it allows the calculation of angles and distances on curved surfaces, such as a sphere or ellipsoid. Since meridians converge and points are not aligned as they would be on a flat plane, spherical trigonometry provides the necessary formulas to compute correct azimuths, distances, and corrections for meridian convergence.

What is the significance of the gamma (γ) angle in meridian convergence?

The gamma (γ) angle represents the additional small angle difference between the direct and reverse azimuths due to the convergence of meridians. It arises because meridians are not parallel on a spherical or ellipsoidal surface. This correction is essential for accurate azimuth calculations when working with non-planar datums.

How do you compute the gamma (γ) angle using spherical trigonometry?

The gamma angle can be computed using the sine rule in spherical trigonometry. The formula involves the direct azimuth, the spherical arc between the points, and other geometric parameters related to the spherical triangle formed by the points and the North Pole.

What is the difference between planar and spherical azimuth calculations?

In planar azimuth calculations, the reverse azimuth is simply 180° different from the direct azimuth. However, in spherical or ellipsoidal datums, due to meridian convergence, the reverse azimuth differs slightly from 180°. The difference is accounted for by the gamma (γ) angle, which is computed using spherical trigonometry.

What role do ellipsoidal datums, such as WGS84, play in azimuth correction due to meridian convergence?

Ellipsoidal datums like WGS84 introduce more complexity in azimuth calculations compared to spherical datums. The geometry of the ellipsoid, including parameters like the radius of the prime vertical and eccentricity, must be factored into the calculations. The correction due to meridian convergence (gamma angle) is more complicated and requires specific formulas to account for the ellipsoidal shape.

What is the formula to compute the reverse azimuth with the meridian convergence correction?

The formula for the reverse azimuth on a spherical or ellipsoidal surface is: Reverse Azimuth = Direct Azimuth + π (or 180°), plus the correction due to meridian convergence (gamma angle). For ellipsoidal surfaces, this involves more complicated terms such as the geodetic distance and eccentricity of the ellipsoid.

Can you provide an example of how to compute the correction due to convergence of meridians?

Yes, in the example provided in the script, the first step is to calculate parameters like the radius of the prime vertical and other ellipsoidal parameters. Using the spherical trigonometry formulas, the gamma (γ) angle correction is computed and converted to degrees, minutes, and seconds. The final correction in this example is 7° 35' 14.43".